Physics 3210 Spring 2008Problem Set 121. For each of the following matrices, find all eigenvalues and a basis for eacheigenspace. Without reference to the geometric multiplicities, determinewhether or not the matrix is dia gonalizable.(a)1 −3 33 −5 36 −6 4(b)−3 1 −1−7 5 −1−6 6 −2[Hint : The eigenvalues are easy to find if you use some judicious elemen-tary row operations.]2. Let A = (aij) be a triangular matrix, and assume that all of the diagonalentries of A are distinct. Is A diagonalizable? Explain.3. Prove that any real symmetric 2 × 2 matrix is diagonalizable.4. Let V be the space of all real polyno mials f ∈ R[x] of degree at most 2,and define T ∈ L(V ) by T f = f + f′+ xf′where f′denotes the usualderivative with respect to x.(a) Write down the most obvious basis { e1, e2, e3} for V you can think of,and then write down [T ]e.(b) Find all eigenvalues of T , and then find a nonsingular matrix P suchthat P−1[T ]eP is diagonal.5. It was shown on page 13 of the LA Part 2 notes that two similar matri-ces A and B have the same eigenvalues. Now you will show that theseeigenvalues also have the same geometric multiplicities.(a) Prove that if P is invertible, then rank(P A) = rank(A). [Hint : Usethe result (see LA 1 pages 63-64) rank(P A) ≤ min{rank(P ), rank(A)},together with the fact that A = P−1(P A).](b) Prove that if Q is invertible, then rank(AQ) = rank(A).(c) Let λ be an eigenvalue of A and B. Prove that dim VAλ= dim VBλ. Inother words, prove dim Ker(λI − A) = dim Ker(λI − B).6. Using geometric multiplicities, determine whether o r not each of the fol-lowing matrices is diagonalizable. If it is, find a nonsingular matrix P anda diagonal matrix D such that P−1AP = D.(a)7 −4 08 −5 06 −6 3(b)0 0 11 0 −10 1 17. Let V be a finite-dimensional inner product space and let T ∈ L(V ) be anormal operator.1(a) Show that if N is any normal operato r on V , then kN xk =N†x.(b) If λ is any scalar, show that the operator N := T − λ1 is normal.(c) Now use the previous two results to show that T x = λx if and only ifT†x = λ∗x.(d) Show that eigenvectors belonging to distinct eigenvalues of a normaloperator are orthogonal.8. Show that if A, B ∈ Mn(F) are unitarily similar and A is normal, then Bis also normal.9. A group (G, ⋄) is a nonempty s e t G together with a binar y operationcalled multiplication and denoted by ⋄ that obeys the following axioms:(G1) a, b ∈ G implies a ⋄ b ∈ G (closure);(G2) a, b, c ∈ G implies (a ⋄ b) ⋄ c = a ⋄ (b ⋄ c) (associativity);(G3) There exists e ∈ G such that a ⋄ e = e ⋄ a = a for all a ∈ G(identity);(G4) For each a ∈ G, there exis ts a−1∈ G such that a⋄a−1= a−1⋄a = e(inverse).(As a side remark, a group is said to be abelian if it also has the propertythat(G5) a ⋄ b = b ⋄ a for all a, b ∈ G (commutativity).In the case o f abelian groups, the gro up multiplication operation is fre-quently denoted by + and called addition. Also, there is no standardnotation for the group multiplication symbol, and my choice of ⋄ is com-pletely arbitrary.)If the number of elements in G is finite, then G is said to be a finite group.I will simplify the notation by leaving out the group multiplication sym-bol and assuming that it is understood for the particular group underdiscussion.Let V be a finite-dimensional inner product space over C, and let G be afinite group. If for each g ∈ G there is a linear operator U (g) ∈ L(V ) suchthatU(g1)U(g2) = U (g1g2)then the collection U (G) = {U(g)} is said to form a representationof G. If W is a subspace of V with the prop erty that U (g)(W ) ⊂ Wfor all g ∈ G, then we say W is U (G)-invariant (or simply invariant).Furthermore, we s ay that the representation U (G) is irreducible if thereis no nontrivial U(G)-invariant subspace (i.e., the only invar iant subspacesare {0} and V itself).(a) Prove Schur’s lemma 1: Let U (G) be an irreducible representationof G on V . If A ∈ L(V ) is such that AU (g) = U (g)A for all g ∈ G,then A = λ1 where λ ∈ C. [Hint : Let λ be an eigenvalue of A withcorresponding eigenspace Vλ. Show that Vλis U(G)-invariant.]2(b) If S ∈ L(V ) is nonsingular, show that U′(G) = S−1U(G)S is also arepresentation of G on V . (Two repres e ntations of G related by sucha similarity transformation are said to be equivalent.)(c) Prove Schur’s l emma 2: Let U (G) and U′(G) be two irreducible rep-resentations of G on V and V′respectively, and suppose A ∈ L(V′, V )is such that AU′(g) = U(g)A for all g ∈ G. Then either A = 0, orelse A is an isomorphism of V′onto V so that A−1exists and U (G) isequivalent to U′(G). [Hint : Show that Im A is invaria nt under U (G),and that Ker A is invariant under